《航空器的稳定与控制》（英文版）Lecture 8 Aircraft Lateral dynamics

16.333 Lecture #8 Aircraft Lateral Dynamics Spiral, Roll, and dutch roll Modes

16.333 Lecture # 8 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes

Fa2004 16.3337-1 Aircraft Lateral dynamics Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics Ax+ Bu T d yb= rsec go m g cos o 4=(+M)(+层N)(+层N)0 (ILLr+ tan e ere xxZ xxz I=Iax/(xI-盈) and 0 B 0

� � � � Fall 2004 16.333 7–1 Aircraft Lateral Dynamics • Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics ⎤⎡ v ⎥ ⎥ ⎥ ⎦ δa , u = δr x˙ = Ax + Bu , x = ⎢ ⎢ ⎢ ⎣ p r φ and ψ˙ = r sec θ0 ⎡ ⎤ A = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Yv Yp Yr m m m − U0 g cos θ0 ( I L � v + I� Nv) ( Lp + I� Np) (Lr + I� Nr) 0 zx zx I� zx I� xx xx xx (I� Lv + Nv ) (I� Lp + Np ) (I� Lr + Nr ) 0 zx I� zx zx I� I� zz zz zz 0 1 tan θ0 0 where I� = (IxxIzz − I2 xx zx)/Izz I� = (IxxIzz − I2 zz zx)/Ixx I� = Izx/(IxxIzz − I2 zx zx) and ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ (m) −1 0 0 0 (I� xx) −1 I� zx ⎥ ⎥ ⎥ ⎦ ⎡ ⎤ Yδa Yδr ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ B = ⎣ L ⎦ δa Lδr zz) −1 · zx 0 I (I Nδa Nδr 0 0 0

Fa2004 16.3337-2 Lateral stability Derivatives a key to understanding the lateral dynamics is roll-yaw coupling Lp rolling moment due to roll rate Roll rate p causes right to move wing down, left wing to move up Vertical velocity distribution over the wing W=py Leads to a spanwise change in the AoA ar( y)=py/Uo Creates lift distribution(chordwise strips) SLu(y=spuoCl ar(y)cydy Net result is higher lift on right, lower on left Rolling moment b/2 L SLu(3) (y)dy=-DpUb Cla tra-cydy= Lp0 Key point: positive yaw rate= positive roll moment

� � � � Fall 2004 16.333 7–2 Lateral Stability Derivatives • A key to understanding the lateral dynamics is roll­yaw coupling. • Lp rolling moment due to roll rate: – Roll rate p causes right to move wing down, left wing to move up → Vertical velocity distribution over the wing W = py – Leads to a spanwise change in the AOA: αr(y) = py/U0 – Creates lift distribution (chordwise strips) 1 δLw(y) = ρU0 2 Clααr(y)cydy 2 – Net result is higher lift on right, lower on left – Rolling moment: b/2 b/2 L = δLw(y)·(−y)dy = −2 1 ρU0 2 −b/2 Clα py2 cydy ⇒ Lp 0 6 4 – Key point: positive yaw rate ⇒ positive roll moment

Fa2004 16.3337-3 p yawing moment due to roll rate Rolling wing induces a change in spanwise Aoa, which changes the spanwise lift and drag Distributed drag change creates a yawing moment. Expect higher drag on right(lower on left )- positive yaw moment There is both a change in the lift(larger on downward wing be- cause of the increase in a)and a rotation(leans forward on down- ward wing because of the larger a).- negative yaw moment In general hard to know which effect is larger. Nelson suggests at for a rectangular wing, crude estimate is that Nn≈pUSb(-)0<0

Fall 2004 16.333 7–3 • Np yawing moment due to roll rate: – Rolling wing induces a change in spanwise AOA, which changes the spanwise lift and drag. – Distributed drag change creates a yawing moment. Expect higher drag on right (lower on left) → positive yaw moment – There is both a change in the lift (larger on downward wing be￾cause of the increase in α) and a rotation (leans forward on down￾ward wing because of the larger α). → negative yaw moment – In general hard to know which effect is larger. Nelson suggests that for a rectangular wing, crude estimate is that 1 Np ≈ ρU0 2 Sb(−CL) 0 < 0

Fa2004 16.3337-4 Numerical results The code gives the numerical values for all of the stability derivatives Can solve for the eigenvalues of the matrix a to find the modes of the system 0.0331±0.9470i -0.5633 -0.0073 Stable, but there is one very slow pole There are 3 modes, but they are a lot more complicated than the longitudinal case ow mo 0.0073 Spiral Mode Fast real 0.5633 Roll Damping Oscillatory-0.0331±0.9470→ Dutch roll Can look at normalized eigenvectors Spiral Roll Dutch Rol B=/00 00067001970.3269-28° p=p/(20/6b)-000907120.1992° 个=7/(20/b)000520004000368-112 1000100010000° Not as enlightening as the longitudinal case

Fall 2004 16.333 7–4 Numerical Results • The code gives the numerical values for all of the stability derivatives. Can solve for the eigenvalues of the matrix A to find the modes of the system. −0.0331 ± 0.9470i −0.5633 −0.0073 – Stable, but there is one very slow pole. • There are 3 modes, but they are a lot more complicated than the longitudinal case. Slow mode ­0.0073 ⇒ Spiral Mode Fast real ­0.5633 ⇒ Roll Damping Oscillatory −0.0331 ± 0.9470i ⇒ Dutch Roll Can look at normalized eigenvectors: Spiral Roll Dutch Roll β = w/U0 0.0067 ­0.0197 0.3269 ­28◦ pˆ = p/(2U0/b) ­0.0009 ­0.0712 0.1198 92◦ rˆ = r/(2U0/b) 0.0052 0.0040 0.0368 ­112◦ φ 1.0000 1.0000 1.0000 0◦ Not as enlightening as the longitudinal case

Fa2004 16.3337-5 Lateral modes Roll Damping -well damped As the plane rolls, the wing going down has an increased a (wind is effectively"coming up"more at the wing Opposite effect for other wing There is a difference in the lift generated by both wings more on side going down The differential lift creates a moment that tends to restore the equilibrium. Recall that Lp <0 After a disturbance, the roll rate builds up exponentially until the restoring moment balances the disturbing moment, and a steady roll is established Disturbing rolling moment Restoring rolling moment Port wing Starboard Reduction in incidence

Fall 2004 16.333 7–5 Lateral Modes Roll Damping ­ well damped. – As the plane rolls, the wing going down has an increased α (wind is effectively “coming up” more at the wing) – Opposite effect for other wing. – There is a difference in the lift generated by both wings → more on side going down – The differential lift creates a moment that tends to restore the equilibrium. Recall that Lp < 0 – After a disturbance, the roll rate builds up exponentially until the restoring moment balances the disturbing moment, and a steady roll is established. py V0 -py − �' �' Roll Rate p Disturbing rolling moment Restoring rolling moment V0 Port wing Starboard wing Reduction in incidence Reduction in incidence

Fa2004 16.3337-6 Spiral Mode-slow, often unstable From level flight consider a disturbance that creates a small rol angle >0- This results in a small side-slip v(vehicle slides own Now the tail fin hits on the oncoming air at an incidence angle b extra tail lift- positive yawing moment Moment creates positive yaw rate that creates positive roll mo- ment (Lr>0) that increases the roll angle and tends to increase he side-slip → makes things worse If unstable and left unchecked, the aircraft would fly a slowly diverging path in roll, yaw, and altitude= it would tend to spiral into the ground Sideslip Steadily increasing roll angle Fin lift disturbance B/ Steadily increasing yaw Yawing moment due to fin lift z Fin lift force o Can get a restoring torque from the wing dihedral Want a small tail to reduce the impact of the spiral mode

Fall 2004 16.333 7–6 Spiral Mode ­ slow, often unstable. – From level flight, consider a disturbance that creates a small roll angle φ > 0 → This results in a small side­slip v (vehicle slides downhill) – Now the tail fin hits on the oncoming air at an incidence angle β → extra tail lift → positive yawing moment – Moment creates positive yaw rate that creates positive roll mo￾ment (Lr > 0) that increases the roll angle and tends to increase the side­slip → makes things worse. – If unstable and left unchecked, the aircraft would fly a slowly diverging path in roll, yaw, and altitude ⇒ it would tend to spiral into the ground!! • Can get a restoring torque from the wing dihedral • Want a small tail to reduce the impact of the spiral mode

Fa2004 16.3337-7 Dutch Roll -damped oscillation in yaw, that couples into roll Frequency similar to longitudinal short period mode, not as well damped( fin less effective than horizontal tail) Consider a disturbance from straight-level flight Oscillation in yaw al(fin provides the aerodynamic stiffness) o Wings moving back and forth due to yaw motion result in oscil- latory differential lift drag(wing moving forward generates more lift)Lr>0 -Oscillation in roll that lags y by approximately 90 Forward going wing is low Oscillating roll = sideslip in direction of low wing !(a) 方> Path traced by starboard wing tip in one dutch roll cyo (a) Starboard wing yaws aft with aft yaw angle as aircraft rolls through (d) starboard wing reaches maximum orward yaw angle as aircraft rolls through wings level in negative sense to zero with positive damping

Fall 2004 16.333 7–7 Dutch Roll ­ damped oscillation in yaw, that couples into roll. • Frequency similar to longitudinal short period mode, not as well damped (fin less effective than horizontal tail). • Consider a disturbance from straight­level flight → Oscillation in yaw ψ (fin provides the aerodynamic stiffness) → Wings moving back and forth due to yaw motion result in oscil￾latory differential lift/drag (wing moving forward generates more lift) Lr > 0 → Oscillation in roll φ that lags ψ by approximately 90◦ ⇒ Forward going wing is low Oscillating roll ⇒ sideslip in direction of low wing

Fa2004 16.3337-8 Do you know the origins on the name of the mode? Damp the dutch roll mode with a large tail fin

Fall 2004 16.333 7–8 • Do you know the origins on the name of the mode? • Damp the Dutch roll mode with a large tail fin

Fa2004 16.3337-9 Aircraft Actuator Influence g8 ePpl 品品>95E=5专 Figure 1: Aileron impulse to flight variables. Response primarily in g Transfer functions dominated by lightly damped Dutch-roll Note the rudder is physically quite high, so it also influences the a/c roll Ailerons influence the Yaw because of the differential drag

Fall 2004 16.333 7–9 Aircraft Actuator Influence 10−2 10−1 100 10−2 10−1 100 101 102 |G adβ | Freq (rad/sec) 10−2 10−1 100 10−2 10−1 100 101 102 |G adp | Freq (rad/sec) Transfer function from aileron to flight variables 10−2 10−1 100 10−2 10−1 100 101 102 |G adr | Freq (rad/sec) 10−2 10−1 100 −100 −150 −200 −50 0 50 200 150 100 arg G adβ Freq (rad/sec) 10−2 10−1 100 −100 −150 −200 −250 −300 −350 −50 0 arg G adp Freq (rad/sec) 10−2 10−1 100 −100 −150 −200 −50 0 50 200 150 100 arg G adr Freq (rad/sec) Figure 1: Aileron impulse to flight variables. Response primarily in φ. • Transfer functions dominated by lightly damped Dutch­roll mode. • Note the rudder is physically quite high, so it also influences the A/C roll. • Ailerons influence the Yaw because of the differential drag